It is important to note that pairwise independence does not imply mutual independence, as illustrated by Example 3.2.5.

example 3.2.5 Suppose X and Y are independent random variables which each take values 1 or —1 with probability 0.5 and define Z = XY. Then Z is independent of either X or Y, but X, Y, and Z are not mutually independent because

example 3.2.6 Let the sample space S be the set of eight integers 1 through 8 with the equal probability of У8 assigned to each of the eight integers. Find three random variables (real-valued functions) defined over S which are mutually independent.